We derive simple analytical expressions for the particle density rho(r) and the kinetic energy density tau(r) for a system of noninteracting fermions in a d-dimensional isotropic harmonic oscillator potential. We test the Thomas-Fermi (TF, or local-density) approximation for the functional relation tau[rho] using the exact rho(r) and show that it locally reproduces the exact kinetic energy density tau(r), including the shell oscillations, surprisingly well everywhere except near the classical turning point. For the special case of two dimensions (2D), we obtain the unexpected analytical result that the integral of tau(TF)[rho(r)] yields the exact total kinetic energy.

We present closed analytical expressions for the particle and kinetic energy spatial densities at finite temperatures for a system of noninteracting fermions (bosons) trapped in a d-dimensional harmonic oscillator potential. For d=2 and 3, exact expressions for the N-particle densities are used to calculate perturbatively the temperature dependence of the splittings of the energy levels in a given shell due to a very weak interparticle interaction in a dilute Fermi gas. In two dimensions, we obtain analytically the surprising result that the |l|-degeneracy in a harmonic oscillator shell is not lifted in the lowest order even when the exact, rather than the Thomas-Fermi expression for the particle density is used. We also demonstrate rigorously (in two dimensions) the reduction of the exact zero-temperature fermionic expressions to the Thomas-Fermi form in the large-N limit.Comment: 14 pages, 4 figures include

Using two distinct inversion techniques, the local one-dimensional potentials for the Riemann zeros and prime number sequence are reconstructed. We establish that both inversion techniques, when applied to the same set of levels, lead to the same fractal potential. This provides numerical evidence that the potential obtained by inversion of a set of energy levels is unique in one-dimension. We also investigate the fractal properties of the reconstructed potentials and estimate the fractal dimensions to be D = 1.5 for the Riemann zeros and D = 1.8 for the prime numbers. This result is somewhat surprising since the nearest-neighbour spacings of the Riemann zeros are known to be chaotically distributed whereas the primes obey almost poisson-like statistics. Our findings show that the fractal dimension is dependent on both the level-statistics and spectral rigidity, ∆3, of the energy levels.

We consider two bosonic atoms interacting with a short-range potential and trapped in a spherically symmetric harmonic oscillator. The problem is exactly solvable and is relevant for the study of ultra-cold atoms. We show that the energy spectrum is universal, irrespective of the shape of the interaction potential, provided its range is much smaller than the oscillator length.

We systematically develop a density functional description for the equilibrium properties of a two-dimensional, harmonically trapped, spin-polarized dipolar Fermi gas based on the Thomas-Fermi von Weizsäcker approximation. We pay particular attention to the construction of the twodimensional kinetic energy functional, where corrections beyond the local density approximation must be motivated with care. We also present an intuitive derivation of the interaction energy functional associated with the dipolar interactions, and provide physical insight into why it can be represented as a local functional. Finally, a simple, and highly efficient self-consistent numerical procedure is developed to determine the equilibrium density of the system for a range of dipole interaction strengths. PACS numbers: 31.15.E-, 71.10.Ca, 03.75.Ss, 05.30.Fk I. INTRODUCTIONUltra-cold, trapped dipolar quantum gases have received increasing attention over the past decade owing to the inherently interesting properties of the anisotropic, and long-range nature of the dipole-dipole interaction [1]. One of the important consequences of the anisotropy is that the interactions between the particles can be tuned from being predominantly attractive to repulsive by simply changing the 3D trapping geometry, or for dipoles confined to the 2D x-y plane, by adjusting the orientation of the dipoles relative to the z-axis [1, 2]. Therefore, novel physics in both the equilibrium and dynamic properties of such systems may be explored as a function of the strength of the interaction, the geometry of the confining potential, and the dimensionality of the system.While the degenerate dipolar Bose gas has been well studied experimentally and theoretically [1], realizing a degenerate dipolar Fermi gas in the laboratory has proven to be much more elusive. One of the reasons for this is that the path to quantum degeneracy is impeded by the Pauli principle, which forbids s-wave collisions between identical atoms. Thus, early attempts to cool both magnetic and molecular dipolar Fermi gases below degeneracy were unsuccessful [3][4][5][6]. However, in the recent work of M. Lu et al. [7], this experimental hurdle was finally overcome, resulting in the experimental realization of a spin polarized, degenerate dipolar Fermi gas. Specifically, using the method of sympathetic cooling, a mixture consisting of 161 Dy and the bosonic isotope 162 Dy, were cooled to T /T F ∼ 0.2. In addition, this group was also able to evaporatively cool a single component gas of 161 Dy down to a temperature of T /T F ∼ 0.7. This latter result is presumed to arise from the rethermalization provided by the strong dipolar scattering between the 161 Dy atoms which have a large magnetic moment (µ ∼ 10µ B ).The ability to fabricate such systems in the laboratory now opens the door for the investigation of both the equilibrium and dynamical properties of dipolar Fermi gases, and will enable contact to be made with the large body of theoretical work already in the literature [1]. Moreover, it is now r...